含夹杂双重孔隙介质中散射问题的研究
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摘要
近年来，有关双重孔隙介质波动理论和相关问题的研究越来越多的得到人们的关注，尤其是双重孔隙介质中夹杂对弹性波的散射问题一直是科学界具有挑战性的研究课题，在物理学、地震学、力学中都有着广泛而深远的意义。例如复合材料无损探测、矿产和石油的勘探、雷达、声纳等技术的应用，均需要考虑弹性波的散射效应与复杂孔隙介质的力学特性之间的相互关系。
迄今为止，尽管众多的学者对弹性介质以及单孔隙介质中夹杂对弹性波的散射问题做了许多卓有成效的研究工作，但双重孔隙介质中的相关问题还少有研究，需要更深入的探讨，而双重孔隙介质中弹性波的散射问题的研究则更是处于起步阶段，所以本文主要针对双重孔隙介质中含球形夹杂的散射问题进行了研究，基于Berryman等建立的拓展Biot理论和双重孔隙介质模型，本文将波函数展开法推广应用到求解平面和三维两种状态下弹性波在双重孔隙介质中单球形理想流体夹杂上的散射问题。文中给出了问题解析解的形式，包括位移场和应力场。进一步数值计算获得了P1~P3、S波以及三维情况下SH波和SV波散射场的谐波振幅随频率的变化规律。本文的计算结果表明，基质孔隙率、裂隙孔隙率、夹杂半径和基质渗透率均对压缩波和剪切波散射场的谐波振幅有显著的影响，并且随着频率的增大，基质孔隙率、裂隙孔隙率、夹杂半径和基质渗透率对散射场谐波振幅的影响程度均增高。但基质孔隙率对P3波的谐波振幅基本都没有影响，而裂隙孔隙率对P2波的谐波振幅也影响甚微，基质渗透率只对P2波的散射场谐波振幅有影响，而对其他的波的散射场谐波振幅基本没有影响。在不同基质孔隙率、裂隙孔隙率、夹杂半径和基质渗透率变化下，平面散射问题的压缩波、S波与三维散射问题相应的压缩波、SH波散射场谐波振幅随频率的变化规律都相同。在同一频率下，平面散射问题的压缩波、S波与三维散射问题相应的压缩波、SH波散射场谐波振幅随基质孔隙率、裂隙孔隙率、夹杂半径和基质渗透率变化的比例也相近。
For the past few years, the research on the theory of elastic wave in double-porosity mediumare paid more and more attention by scholars, especially the problem that scattering of elastic waveby inclusion in double-porosity medium is always a challenging subject in scientific community, ithas a far-reaching significance in both physics, seismology and mechanics. Many situation inengineering have to take the correlations of scattering effect and mechanics properties of complexporous medium into account, for example, nondestructive testing of composite material, mineralprospecting and oil exploration, technique of radar and sonar.
Although there are many scholars did a fair amount of effective work on the problem thatscattering of elastic wave by inclusion in homogeneously elastic medium and single-porositymedium so far, there are few studies about double-porosity medium, especially the scattering ofelastic wave by inclusion in double-porosity medium. In this paper, base on the extended Biot'stheory of poroelasticity and double-porosity model which founded by Berryman and Wang, byusing the wave expansion method, the scattering of longitudinal wave by a spherical ideal fluidinclusion in double-porosity medium is investigated. An explicit analytical solutions for theharmonic amplitude, displacement field and stress field of the problem were obtained. Then thepaper investigated the spherical harmonic amplitude of P1~P3wave, S wave in plane problem andSH wave, SV wave in three-dimension problem in the scattered field by numerical analysis, andsummarized the rules. The results of the calculation show that matrix porosity, fracture porosity,radius of inclusion and matrix permeability have obvious influence on spherical harmonic amplitudeof scattered wavefield. And the influence of matrix porosity, fracture porosity, radius of inclusionand matrix permeability would become more obviously while the frequency increase. But the matrixporosity has no influence on the spherical harmonics amplitude of P3wave scattered field, and fracture porosity has little influence on the spherical harmonic amplitude of P2wave scattered fieldboth in the plane problem and the three-dimension problem. Matrix permeability only has influenceon spherical harmonic amplitude of P2wave scattered field, in other words, matrix permeability hasno influence on the spherical harmonics amplitude of P1waves scattered field, P3wave scatteredfield, S wave scattered field in plane problem and SH wave scattered field, SV wave scattered fieldin three-dimension problem. The compressional waves in the plane scattering problem have thesame rules as the corresponding compressional waves in the three-dimension scattering problem,under different matrix porosity, fracture porosity, radius of inclusion or matrix permeability. The Swave in the plane scattering problem and the SH wave in the three-dimension scattering problemhave the similar relationship. As the change of matrix porosity, fracture porosity, radius of inclusionor matrix permeability under the same frequency, the compressional waves in the plane scatteringproblem have the same change scale with the corresponding compressional waves in thethree-dimension scattering problem. The S wave in the plane scattering problem and the SH wave inthe three-dimension scattering problem also have the similar relationship.
迄今为止，尽管众多的学者对弹性介质以及单孔隙介质中夹杂对弹性波的散射问题做了许多卓有成效的研究工作，但双重孔隙介质中的相关问题还少有研究，需要更深入的探讨，而双重孔隙介质中弹性波的散射问题的研究则更是处于起步阶段，所以本文主要针对双重孔隙介质中含球形夹杂的散射问题进行了研究，基于Berryman等建立的拓展Biot理论和双重孔隙介质模型，本文将波函数展开法推广应用到求解平面和三维两种状态下弹性波在双重孔隙介质中单球形理想流体夹杂上的散射问题。文中给出了问题解析解的形式，包括位移场和应力场。进一步数值计算获得了P1~P3、S波以及三维情况下SH波和SV波散射场的谐波振幅随频率的变化规律。本文的计算结果表明，基质孔隙率、裂隙孔隙率、夹杂半径和基质渗透率均对压缩波和剪切波散射场的谐波振幅有显著的影响，并且随着频率的增大，基质孔隙率、裂隙孔隙率、夹杂半径和基质渗透率对散射场谐波振幅的影响程度均增高。但基质孔隙率对P3波的谐波振幅基本都没有影响，而裂隙孔隙率对P2波的谐波振幅也影响甚微，基质渗透率只对P2波的散射场谐波振幅有影响，而对其他的波的散射场谐波振幅基本没有影响。在不同基质孔隙率、裂隙孔隙率、夹杂半径和基质渗透率变化下，平面散射问题的压缩波、S波与三维散射问题相应的压缩波、SH波散射场谐波振幅随频率的变化规律都相同。在同一频率下，平面散射问题的压缩波、S波与三维散射问题相应的压缩波、SH波散射场谐波振幅随基质孔隙率、裂隙孔隙率、夹杂半径和基质渗透率变化的比例也相近。
For the past few years, the research on the theory of elastic wave in double-porosity mediumare paid more and more attention by scholars, especially the problem that scattering of elastic waveby inclusion in double-porosity medium is always a challenging subject in scientific community, ithas a far-reaching significance in both physics, seismology and mechanics. Many situation inengineering have to take the correlations of scattering effect and mechanics properties of complexporous medium into account, for example, nondestructive testing of composite material, mineralprospecting and oil exploration, technique of radar and sonar.
Although there are many scholars did a fair amount of effective work on the problem thatscattering of elastic wave by inclusion in homogeneously elastic medium and single-porositymedium so far, there are few studies about double-porosity medium, especially the scattering ofelastic wave by inclusion in double-porosity medium. In this paper, base on the extended Biot'stheory of poroelasticity and double-porosity model which founded by Berryman and Wang, byusing the wave expansion method, the scattering of longitudinal wave by a spherical ideal fluidinclusion in double-porosity medium is investigated. An explicit analytical solutions for theharmonic amplitude, displacement field and stress field of the problem were obtained. Then thepaper investigated the spherical harmonic amplitude of P1~P3wave, S wave in plane problem andSH wave, SV wave in three-dimension problem in the scattered field by numerical analysis, andsummarized the rules. The results of the calculation show that matrix porosity, fracture porosity,radius of inclusion and matrix permeability have obvious influence on spherical harmonic amplitudeof scattered wavefield. And the influence of matrix porosity, fracture porosity, radius of inclusionand matrix permeability would become more obviously while the frequency increase. But the matrixporosity has no influence on the spherical harmonics amplitude of P3wave scattered field, and fracture porosity has little influence on the spherical harmonic amplitude of P2wave scattered fieldboth in the plane problem and the three-dimension problem. Matrix permeability only has influenceon spherical harmonic amplitude of P2wave scattered field, in other words, matrix permeability hasno influence on the spherical harmonics amplitude of P1waves scattered field, P3wave scatteredfield, S wave scattered field in plane problem and SH wave scattered field, SV wave scattered fieldin three-dimension problem. The compressional waves in the plane scattering problem have thesame rules as the corresponding compressional waves in the three-dimension scattering problem,under different matrix porosity, fracture porosity, radius of inclusion or matrix permeability. The Swave in the plane scattering problem and the SH wave in the three-dimension scattering problemhave the similar relationship. As the change of matrix porosity, fracture porosity, radius of inclusionor matrix permeability under the same frequency, the compressional waves in the plane scatteringproblem have the same change scale with the corresponding compressional waves in thethree-dimension scattering problem. The S wave in the plane scattering problem and the SH wave inthe three-dimension scattering problem also have the similar relationship.
引文
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[2] Biot, M. A., Theory of propagation of elastic waves in a fluid-saturated porous solid.Ⅰ.Low-frequency range [J]. J.Acoust. Soc.Am.,1956,28(2):168-178
[3] Biot, M. A., Theory of propagation of elastic waves in a fluid-saturated porous solid.Ⅱ.High-frequency range [J]. J.Acoust. Soc.Am.,1956,28(2):179-191
[4] Biot, M. A., The elasyic coefficients of the theory of consolidation [C]. Applied MechanicsDivision Summer Conference,1957,24:594-601
[5] Biot, M. A., Generalized theory of acoustic propagation in porous dissipative media [J]. J.Acoust. Soc. Am.,1962,34(9):1254-1264
[6] Biot, M. A., Mechanics of deformation and acoustic propagation in porous media [J]. J. Appl.Phys.,1962,33(4):1482-1498
[7] J. W. Strutt, Investigation of the disturbance produced by a spherical obstacle on the waves ofsound [J]. Proceedings of London Mathematical Society v4,1872:253-283
[8] K. Sczawa, Scattering of elastic waves and some allied problems [J]. Bu11. Earthquake Res.Inst.Tokyo Uniw.,1927.
[9] Ying, C. F., and Truell, R., Scattering of a plane longitudinal wave by by a spherical obstaclein an isotropically elastic solid [J]. J. Applied. Physics,1956,27(9):1086-1097
[10] Einspruch, N. G., Witterholt, E. J. And Truell, R., Scattering of a plane longitudinal wave by aspherical fluid obstacle in an elastic medium [J]. J. Acoust. Soc. Am.,1960,32(2):214-220
[11] Pao, Y. H., Scattering of plane compressional waves by a spherical obstacle [J]. J. AppliedPhysics [J].1963,34(3):493-499
[12] Liu, Z. P., and Cai, L. W., Three-dimensional multiple scattering of elastic wave by sphericalinclusions [J]. Journal of Vibration and Acoustics,2009,131(6)
[13] J. T. Chen, Y. T. Lee, Y. J. Lin, Analysis of multiple-spheres radiation and scattering problemsby using the null-field integral equation approach [J]. Appl. Acoust.2010,71(8):690-700
[14] Plona, T. J., Observation of a second bulk compressional wave in a porous medium atultrasonic frequencies [J]. Appl. Phys. Lett.,1980,36:259-261
[15] Kelder, O., Smeulders, D. M. J., Observation of the Biot slow wave in water-saturatedNivelsteiner sandstone [J]. Geophysics,1997,62:1794-1796
[16]诸国桢,流体饱和多孔介质中的声波[J].应用声学,1994,13:1-6
[17] Berryman, J. G., Scattering by a spherical inhomogeneity in a fluid-saturated porous medium[J]. J. Math. Phys.,1985,26(6):1408-1419
[18] Yamakawa, N., Scattering and attenuation of elastic waves [J]. Geophys. Mag.,1962,31:63-97
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